Finite Square-Well Potential

Shackleford

24. Apply the boundary conditions to the finite square-well potential at x=0 to find the relationships between the coefficients A, C, and D and the ratio C/D.

I understand the wave equations in the three separate regions. For this question I need to only consider I, II. The wave equations need to decrease to zero as x approaches positive or negative infinity. The wave equation and its derivative need to be continuous as well. Thus, the wave equation of I equals II.

My professor did a similar problem last semester, but I can't make sense of his procedure. I think the delta function is in it, etc.

http://i111.photobucket.com/albums/n...g?t=1283054054

vela

Your professor wrote the two equations that correspond to

Shackleford

He did it a bit differently, though.

I also re-worked my part, too.

http://i111.photobucket.com/albums/n...g?t=1283054054

He has

1 = A + B
K = ik (A - B

ik/K = (A + B)/(A - B)

(ik + K)/(ik - K) = A/B = delta

I understand the manipulation up until here. I still don't know how in the heck this helps me related A, C, and D. Do I do the same thing?

vela

I think you're getting your coefficients mixed up. Your equations should be

A = C+D
αA = ik(C-D)

I'm not sure what your professor is doing. It looks like his A and B are your C and D and his K is your alpha. He took (your) A to be equal to 1 for some reason. The delta is not the delta function. It's just the quantity which equals A/B.

Shackleford

Oops. I accidentally wrote down B + C for some reason.

Here's what the professor did last week. I assume the book is looking for something like this. I don't know how he got this.

http://i111.photobucket.com/albums/n...g?t=1283058414

vela

You have two equations and three unknowns, so you can solve for two of them, say C and D, in terms of the other, A.

That's what your professor did except in his case, there were four unknowns, so he solved for B and C in terms of A and D.

Shackleford

Okay. I'll play around with the equations. Maybe I'll get partial credit. lol.

Oh, for the 1 = A + B, I think he used the free particle solution for the wave heading from the negative x direction towards the potential.

Shackleford

Well, I tried to play around with what the professor did, but I couldn't get his equations. Forgive my lack of algebraic-manipulation skills.

vela

Rewriting the equations a bit, you get

[tex]\begin{align*}
B - C & = -A + D \\
k_0 B + kC &= k_0 A + kD
\end{align*}
[/tex]

Multiply the first equation by k, add it the second, and solve for B.

Shackleford

Okay. That makes sense. I'm still not sure about my problem. Playing with it, I got

A = [(ik + α)C + (α - ik)D] / 2α

Is that what the book is looking for?

vela

Probably not. Try solving for C and D in terms of A. Then you can calculate the ratio C/D.

Shackleford

Crap. I forgot about the ratio C/D. Well, maybe I'll get partial credit. The homework was due today.

Just now, I got A = [-2ik/(α-ik)] D.

I suspect C would be something similar.

vela

I think that's right, and C comes out similarly.

Shackleford

Well, that's just great. I do something right after it's due. I'm not going to do well on this problem set. Pretty much the rest of the problem sets will come from this book. I don't expect to do well on any of them.

http://www.amazon.com/Quantum-Physic.../dp/0471057002

vela

If you don't like that text, you might want to see if your library has Griffith's book on quantum mechanics. I'll admit I've never seen it, but I used his particle physics book as an undergrad. He was very good at explaining concepts and showing how to apply them in problems.

http://www.amazon.com/Introduction-Q...ref=pd_sim_b_1

(It's the top-selling book on quantum mechanics at Amazon.)

Shackleford

You think it'll help me do the Gasiorowicz problems? Even the professor said Gasiorowicz isn't an ideal textbook. Also, I saw one of my friend's particle physics book, and it's Griffiths, too.

vela

Yes, I think it would help. It couldn't hurt. Textbooks are so expensive, though, so I'd try to look through a copy in a bookstore or at the library first.